Eikonal method in position space

Given the power of the eikonal method as described in the previous subsection, we take the opportunity to offer a different perspective on some of the calculations above. In particular, we would like to understand the eikonal method in position space and in relation to perturbative approaches such as the Schwarzian theory of reparametrization modes.

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In the low-energy SYK model we can perform calculations using the Schwarzian the-ory [12–14,52]. The six-point function is then given by an integral over reparametrization fields ˆt(ˆu):¹⁵

F₆ =N

Z dˆt[ˆu]

SL(2,R) e^iS[ˆt(ˆu)]G_∆1(ˆu1,uˆ2)G_∆j(ˆu3,uˆ4)G_∆2(ˆu5,uˆ6), (5.9) where the normalization N is the inverse of the same integral, but without the eikonal phase inserted — this corresponds to the product of two-point functions as in (2.3). The Schwarzian action (in Lorentzian signature) is given by

iS[ˆt(ˆu)] =−iC

Z dˆu{ˆt(ˆu),u}ˆ , (5.10) where the integral runs over real time ˆu along contours such as those shown in figures 2 or5. The reparametrization bilocals take the form of reparametrized conformal two-point functions:

G_∆(ˆua,uˆ_b) =

" −ˆt⁰(ˆub)ˆt⁰(ˆub) (ˆt(ˆu_a)−ˆt(ˆu_b))²

#∆

(5.11) We now make the following assumption: instead of doing the integral over arbitrary reparametrizations (modulo SL(2,R)), we can approximate the effect of the operator inser-tions by means of finite SL(2,R) transformations with fixed points at asymptotic locations ˆ

u → ±∞. This is implemented through the following piecewise transformation (cf., [52]

for a similar calculation in the case of four-point functions¹⁶):

ˆt(ˆu) =x− (1 +x)²X⁻

4 + (1 +x)X⁻θ(2,3) + (1−x)²X⁺

4 + (1−x)X⁺ θ(5,6) + (1−x)²Y⁺

4 + (1−x)Y⁺θ(3) + (· · ·)θ(4)− (1 +x)²Y⁻

4 + (1 +x)Y⁻θ(5),

(5.12)

wherex= tanh^u₂^ˆ. Theθ-functions indicate the contours on which the respective SL(2,R) transformations have support (cf., figure 11). The explicit form of the term on the fourth contour will not play any role. Because (5.12) isnotan SL(2,R) transformation on contours 3 and 5, the Schwarzian action evaluated on this piecewise reparametrization is nontrivial:

iS[ˆt(ˆu)] = iC

2 (X⁺Y⁻+X⁻Y⁺) +. . . (5.13) where we work perturbatively inX^±andY^±(subsequent terms are parametrically smaller in ∼ _C¹). The bilocal operators (5.11) evaluated on the piecewise transformation ˆt(ˆu) become precisely the Fourier transforms of the momentum space wavefunctions, (A.4) and (A.5):

G_∆1(ˆu₁,uˆ₂) =W₁(ˆu₂)e^{−iY−Pˆ−}W₁(ˆu₁) G_∆j(ˆu3,uˆ4) =^DO_j(ˆu3)e^iX+Pˆ+ e^iX−Pˆ−O_j(ˆu4)^E G_∆2(ˆu₅,uˆ₆) =W₂(ˆu₅)e^iY+Pˆ+W₂(ˆu₆)

(5.14)

15Since we will not be computing any quantum corrections, we can ignore the fact that the Schwarzian partition function has a nontrivial measure [57].

16See also [58] for a related approach.

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Similarly, the Schwarzian integral (5.9) becomes an integral over the piecewise SL(2,R) parameters alone, which coincides with the eikonal result (5.8):

F₆=^Z dX⁺dX⁻dY⁺dY⁻e^{iC2 (X+Y−+X−Y+)}G_∆1(ˆu1,uˆ2)G_∆j(ˆu3,uˆ4)G_∆2(ˆu5,uˆ6). (5.15) 5.3 The Lorentzian six-point function F₆ for general insertion times

Here we take the opportunity to discuss F₆(t₁, t₂;a, b) for general a, b (see, e.g., figures 2 and 5). We first discuss the properties of this six-point OTOC and then give its explicit expression resulting from the gravity calculations.

5.3.1 OTOCology: general properties of the correlator

Let us study the general properties of F₆(t₁, t₂;a, b) as in (5.1) for general a, b. It has several interesting features as a Lorentzian quantum field theory observable:

• As long ast1,2 <0,a > t1andb > t2, it ismaximally out-of-time-order, meaning that it cannot be represented on a complex time contour with fewer than three forward-backward segments (which is the maximal number of switchbacks that can occur for six-point functions without introducing redundancies [59,60]).

• For a < −t₂ orb < −t₁ (as in figure 5), F₆ is not irreducibly out-of-time-order: it receives some ‘disconnected’ four-point contributions, which are out-of-time-order by themselves. In this case, F₆ is the OTOC that has been previously discussed in the context of quantum chaos [24,30,56].¹⁷

• For a > −t₂ and b > −t₁ (as in figure 2), removing any pair of identical operators from F₆ leaves behind a four-point function which is in-time-order (i.e., requires only a single switchback in time). The characteristic exponential decay of F₆ will therefore originate entirely from its ‘connected’ properties as a six-point function and cannot be attributed to ‘disconnected’ four-point contributions. This elucidates the structure (1.3). We shall refer to this property by saying that F₆ is irreducibly out-of-time-order. This distinguishes our observable from six-point functions that have been considered previously in the literature.

• Irrespective of the values of real times, the operator ordering along the complex time contour is always maximally braided [24]. This refers to the fact that the pairwise identical operators are all interlinked in Euclidean time, i.e., their supports along contours overlap (as shown by colored lines in figure 2).

17Note the following interesting feature of the six-point function relevant to the traversable wormhole setup (figure 8): all of its four-point factors are OTOCs even though the full six-point function is not maximally out-of-time-order.

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5.3.2 Result from eikonal saddle point and geodesic calculations

The saddle point approximation to the eikonal calculation (appendix A.1) yields the fol-lowing result for generala, b:

F₆ ≈

A similar result is separately obtained through the geodesic approximation in the corresponding double-shock geometry (appendix C) for generala, b:

F₆^(geodesic)≈ andδSiis the entropy introduced by the respective operator insertion, which scales asG∆i

(see (2.8)). Note that if we ignore the terms proportional to ^δS_Sⁱ but not enhanced bye^−ui, this reduces to (5.16). See also (A.11) and (B.8). In particular, for a, b −t_1,2 and for a=b= 0 this reduces to our previous results (2.7) and (3.7), respectively.

Regarding the gravitational calculation, a few comments are in order. The factoriza-tion property of the F₆ correlator has an alternative interpretation from this geometric perspective. In the geodesic approximation, the correlator decays with the exponential of the geodesic distance between the two O_j probes. Since the state has been perturbed by the W1, W2 insertions, the corresponding geometry is now that of the background with two additional shocks. Consequently, the resultant distance naturally expands into a sum of the background, the independent effects of the W₁, W₂ insertions, and one “connected”

contribution. Such an expansion of distance effects then exponentiates into the factoriza-tion of the resultant correlator. This connected contribufactoriza-tion is due to the fact that gravity is not linear, and it corrects for the fact that the two insertions affect each other.

6 Conclusion

Using out-of-time-order six-point functions, we quantified various properties of collisions behind the horizon of particles sent into the eternal AdS black hole from the two asymp-totic regions. In sections 2,3and 4, we considered three different real-time configurations

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to elucidate different aspects of the collision. At a technical level, we used the eikonal re-summation technique as well as gravitational geodesic computations to obtain the relevant expressions for the six-point functions. A common theme was the factorization property of the eikonal six-point function into four-point factors and a connected contribution, cf., (1.3).

Some unanswered questions are as follows:

• In the setup for diagnosing the collision in the wormhole interior, it appears that the existence of the singularity plays a crucial role by means of prohibiting the meeting of signals that are sent in too late. What will happen for geometries without a singularity, such as charged black holes? In that case, signals can meet no matter how late they are sent in, but they will meet in a region where the dilaton is negative, or in other words, where gravity is repulsive. It will be important to understand this phenomenon in the future as the six-point functions we used are not sensitive to this.

• In the setup for detecting the overlap of two perturbations spreading through a shared quantum circuit, the connected part of the six-point function (3.7) gives an order one contribution. This is qualitatively negligible, but physically interesting as it quantifies the amount by which the perturbations do not spread independently. Is there any way to account for this factor in the circuit model?

• In the traversable wormhole setup, we can only extract the signal after the collision if the collision is mild. It seems that this protocol does not allow us to extract information about a strong collision. Similarly, we only demonstrated how to extract one collision product (the φsignal), but not the other one (the ψsignal). These are deficiencies of our protocol. Are there other protocols to improve the situation?¹⁸

• The factorization property (1.3) of the eikonal six-point function calls for a more detailed understanding. Similar structures have appeared before in two-dimensional CFTs [28, 30, 62], where the eikonalization of Virasoro conformal blocks offers an explanation. It would be interesting to argue for (1.3) in greater generality (perhaps using a perturbative expansion as in appendixD). This would open up the possibility to understand higher-point Virasoro identity blocks in interesting regimes such as the Regge limit or in configurations with several heavy operators (generalizing [63]), which would have many applications. We expect that many of the expressions derived in this paper have a direct analog in Virasoro identity conformal blocks.

Acknowledgments

We thank Ahmed Almheiri, Tarek Anous, Adam Levine, Henry Lin, Juan Maldacena for helpful discussions. F.H. is grateful for support from the DOE grant DE-SC0009988 and from the Paul Dirac and Sivian Funds. A.S. and Y.Z. are supported by the Simons foundation through the It from Qubit Collaboration.

18For recent work in this direction, see [61].

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A Eikonal calculations in the Schwarzian theory

In this appendix we give some details regarding the evaluation of the eikonal integrals (5.8) and a similar expression for the traversable wormhole setup.